Xi-Ping Zhu

Sharp dimension estimates of holomorphic functions and rigidity

Let M n be a complete noncompact Kahler manifold of complex dimension n with nonnegative holomorphic bisectional curvature. Denote by O d (M n ) the space of holomorphic functions of polynomial growth of degree at most d on M n . In this paper we prove that dimcod(M n ) ≤ dim c O [d] (C n ), for all d > 0, with equality for some positive integer d if and only if M n is holomorphically isometric to C n . We also obtain sharp improved dimension estimates when its volume growth is not maximal or its Ricci curvature is positive somewhere.

Lipschitz continuity of harmonic maps between Alexandrov spaces

AbstractIn 1997, Jost (Calc Var PDE 5:1–19, 1997) and Lin (Collection of papers on geometry, analysis and mathematical physics, World Sci. Publ., River Edge, 1997), independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally Hölder continuous. Lin (1997) proposed an open problem: can the Hölder continuity be improved to Lipschitz continuity? J. Jost also asked a similar problem about Lipschitz regularity of harmonic maps between singular spaces [see page 38 in Jost (in: Jost, Kendall, Mosco, Röckner, Sturm (eds) New directions in Dirichlet forms, International Press, Boston, 1998)]. The main theorem of this paper gives a complete resolution to it.